Two angles of a triangle are in the ratio `(4:5)`. If the sum of these angles is equal to the third angle, find the smaller angle of the triangle.

Options

A. `40^@`
B. `50^@`
C. `60^@`
D. `65^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the angles
Let the two angles of the triangle be represented as:
- First angle = \(4x\)
- Second angle = \(5x\)

### Step 2: Set up the equation
According to the problem, the sum of these two angles is equal to the third angle. Let the third angle be \(y\). Therefore, we can write the equation:
\[ y = 4x + 5x \]

### Step 3: Use the triangle angle sum property
The sum of all angles in a triangle is \(180^\circ\). Thus, we can write:
\[ 4x + 5x + y = 180 \]

### Step 4: Substitute for \(y\)
From Step 2, we know that \(y = 4x + 5x\). We can substitute this into the equation:
\[ 4x + 5x + (4x + 5x) = 180 \]

### Step 5: Combine like terms
Now, we combine the terms:
\[ 4x + 5x + 4x + 5x = 180 \]
This simplifies to:
\[ 18x = 180 \]

### Step 6: Solve for \(x\)
Now, we will solve for \(x\):
\[ x = \frac{180}{18} = 10 \]

### Step 7: Find the angles
Now that we have \(x\), we can find the two angles:
- First angle = \(4x = 4 \times 10 = 40^\circ\)
- Second angle = \(5x = 5 \times 10 = 50^\circ\)

### Step 8: Find the third angle
Using the values of the first and second angles, we can find the third angle:
\[ y = 4x + 5x = 40 + 50 = 90^\circ \]

### Conclusion
The angles of the triangle are \(40^\circ\), \(50^\circ\), and \(90^\circ\). The smaller angle of the triangle is:
\[ \text{Smaller angle} = 40^\circ \]

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